An Almost Sure Invariance Principle for the Partial Sums of Infima of Independent Random Variables

Abstract

Let $\{X_n, n \geqslant 1\}$ be a sequence of independent random variables uniformly distributed on the unit interval. Put $X^\ast_n = \inf(X_1, X_2,\cdots, X_n)$ and $S_n = X^\ast_1 + X^\ast_2 + \cdots + X^\ast_n, n \geqslant 2, S_1 = 0$. The aim of this note is to give an almost sure invariance principle for $S_n$. Next we extend the given results to the case when $X_n, n \geqslant 1$, are not uniformly distributed but bounded, and moreover, to sums $\hat{S}_n = X^{(m)}_m + X^{(m)}_{m+1} +\cdots + X^{(m)}_n$, where $X^{(m)}_j$ is the $m$th order statistic of $(X_1, X_2,\cdots, X_j)$.